Geometrical design of blade＇s surface and boundary control of Navier-Stokes equations

In this article a new principle of geometric design for blade＇s surface of an impeller is provided. This is an optimal control problem for the boundary geometric shape of flow and the control variable is the surface of the blade. We give a minimal functional depending on the geometry of the blade＇s surface and such that the flow＇s loss achieves minimum. The existence of the solution of the optimal control problem is proved and the Euler-Lagrange equations for the surface of the blade are derived. In addition, under a new curvilinear coordinate system, the flow domain between the two blades becomes a fixed hexahedron, and the surface as a mapping from a bounded domain in R2 into R3 , is explicitly appearing in the objective functional. The Navier-Stokes equations, which include the mapping in their coefficients, can be computed by using operator splitting algorithm. Furthermore, derivatives of the solution of Navier- Stokes equations with respect to the mapping satisfy linearized Navier-Stokes equations which can be solved by using operator splitting algorithms too. Hence, a conjugate gradient method can be used to solve the optimal control problem.     

AN ASYMPTOTIC ANALYSIS METHOD FOR THE LINEAR SHELL

In this paper, using the formal approach of asymptotic expansion for linear elastic shell we can get each term uk successively. According this method the leading term u0 will be identified by an elliptic boundary value problem, other terms will be obtained by the algebraic operations without solving partial differential equations. We give the variational formulation for the leading term U(x) and construct an approximate solution U^KT(x,ξ):= U(x) Ⅱ1Uξ Ⅱ2Uξ,then we give the estimation.     

THE NAVIER-STOKES EQUATIONS IN STREAM LAYER AND ON STREAM SURFACE AND A DIMENSION SPLIT METHODS

THE NAVIER-STOKES EQUATIONS IN STREAM LAYER AND ON STREAM SURFACE AND A DIMENSION SPLIT METHODS@李开泰$College of Sciences,Xi'an Jiaotong University!Xi'an,710049,China @黄艾香$College of Sciences,Xi'an Jiaotong University!Xi'an,710049,China[14]Il’inAA ,PartlydissipativesemigroupsgeneratedbytheNavierStokessystemontwo dimensionalmanifolds,andtheirattractors[J].RussianAcadSciSbMath,1994,78(1):47-76. [15]LadyzhenskayaOA .Themathematicaltheoryofvisc…